= {x ∈ l2| v(x) + c = 0}, where v: l2→ R, such that v(a) = v(π≤ma)and c∈ R.
Any convex and bounded subset U⊂ is called a local section.
Let WZ= a⊕ π>mWq,S, where ais an open convex subset of π≤ml2. Definition 36. A local section Z is said to be transversal in WZif
WZ∩ Z = Z, WZ= WZ,−∪ Z ∪ WZ,+, where
WZ,−= {x ∈ WZ| v(x) + c < 0}, WZ,+= {x ∈ WZ| v(x) + c > 0}, satisfying the condition
v (f (x))) >0, ∀x ∈ WZ. (A.4)
We will refer to (A.4) as the transversality condition.
We have the following easy lemma.
Lemma 37. Let Z be a local transversal section in WZfor (9) and let N⊂ Wq,Sfor some S > 0.
Assume that there exist t1, t2∈ R, t1< t2, such that the following conditions hold for all x∈ N:
ϕ((t1, t2), x)⊂ WZ, ϕ(t1, x)∈ WZ,− and ϕ(t2, x)∈ WZ,+.
Then, for each x0∈ N, there exists unique tZ(x0) ∈ (t1, t2), such that ϕ (tZ(x0), x0)∈ Z. Also, tZ: N → [t1, t2] is continuous.
Using the above lemma we can define a map PZ1→Z2 : Z1→ Z2 for two transversal local sections Z1and Z2, by
PZ1→Z2(x)= ϕ(tZ2(x), x).
A.4. Representation of h-sets and computer-assisted verification of covering relations with one exit direction
Algorithms for computer-assisted verification of covering relations for finite dimensional maps with arbitrary unstable dimension are presented in [42]. Although they can be directly extended to the case of infinite dimensional systems, for self-consistency of the article we pro-vide here a special case, when h-sets have one exit (nominally unstable) direction, i.e. u(N ) = 1 in Definition2.
Let us fix q > 1. In order to treat the system (8) rigorously on a computer we define a data structure which represents h-sets in l2,q:
type HSet where cNis an invertible affine map
cN
Since we assumed u(N ) = 1 in Definition2, the tail of N is always given by
TN= [−1, 1] ·
B =
=⇒ Nf 1reduces to a finite set of inequalities which must be satisfied. These are:
• πi(B) ⊂ (−1, 1) for i = 2, . . . , m,
• S(B) < S(N1), where S is a positive constant in h-set representation (A.5),
• either π1(Bl) <−1 and π1(Br) >1 or π1(Bl) >1 and π1(Br) <−1.
The two cases in the last condition depend on whether the mapping f changes or not the orien-tation along the exit direction. It is easy to see that
H (t,·) = (1 − t) whether f preserves or not the orientation in the exit direction.
A.5. Technical data
The source code of the C++11 program that realises the computer-assisted proof of Lemma10 is available at [38]. Below we list our choices of some parameters of the algorithms.
• All h-sets that appear in Lemma10are represented as a data structure (A.5) with the constant m = 15.
• We set d = 4 as the order of the Taylor method (Section4.4.3) for rigorous integration of (8).
• High-Order Enclosure (Section4.4.2) with d= 4 acts on m = 11 number of modes.
Verification of covering relation N0
=⇒ Nf 1 requires computation of three images of f — see (A.6). In Lemma10we verify 26 covering relations which means, that we have to check 78 = 3 · 26 inclusions.
We run the program which checks all the inequalities required for covering relations listed in Lemma10on a computer equipped with 64 physical cores (128 threads) Intel(R) Xeon(R) CPU E7-8867 v4 @ 2.40 GHz processors. The program finished after 40 minutes, which is the CPU time needed for the longest integration in N28→1=⇒ NP6 29→1.
The algorithm for rigorous integration of the KS equation is a part of the CAPD library [3].
We tested the program with CAPD version 5.0.59 and the C++11 compiler from gcc-5.2 suite.
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